Towards a statistical mechanics for ecological systems

1. Towards a statistical mechanics for ecological systems Jonathan Potts, Postdoctoral Fellow, University of Alberta, May 2013

2. “My prediction is that the next decade will include an expansion of interest in the problem of the organization and robustness of complex adaptive systems from individuals to the biosphere and the interlinked global socio-economic system. In all such systems, macroscopic patterns emerge from the interactions among large numbers of individual agents, and the challenge is to develop a statistical mechanics for such systems, connecting the microscopic and the macroscopic.” Simon Levin (2012) Towards a marriage of theory and data J Roy Soc Interface

3. From mechanism to pattern Movement

4. From mechanism to pattern Directinteractions

5. From mechanism to pattern Mediated interactions

6. From mechanism to pattern Environmental interactions

7. From mechanism to pattern

8. An invasive species called Homo sapiens

9. An invasive species called Homo sapiens

10. Outline

11. Outline • Modelling animal movement: the “correlated random walk” framework

12. Outline • Modelling animal movement: the “correlated random walk” framework • Adding in environmental interactions: step selection functions

13. Outline • Modelling animal movement: the “correlated random walk” framework • Adding in environmental interactions: step selection functions • Including animal-animal interactions: coupled step selection functions

14. Outline • Modelling animal movement: the “correlated random walk” framework • Adding in environmental interactions: step selection functions • Including animal-animal interactions: coupled step selection functions • Throughout: how do these models help us understand space use phenomena?

15. Movement: correlated random walk

16. Movement: correlated random walk Example step length distribution:

17. Movement: correlated random walk Example step length distribution: Example turning angle distribution:

18. Mathematical formulation Probability of moving to position x given that the animal was previously at position y and arrived there on a trajectory is: where is the step length distribution and the turning angle distribution.

19. Adding environmental interactions

20. Adding environmental interactions A, B, C different habitats. B = worse, A = better, C = best.

21. The step selection function Probability of moving to position x given that the animal was previously at position y and arrived there on a trajectory is: • is the step length distribution, • is the turning angle distribution • is a weighting function • E is information about the environment Fortin D, Beyer HL, Boyce MS, Smith DW, Duchesne T, Mao JS (2005) Wolves influence elk movements: Behavior shapes a trophic cascade in Yellowstone National Park. Ecology 86:1320-1330.

22. Example 1: Amazonian bird flocks • is a function denoting the value of each point in the study area • Use this to test various hypotheses about the nature of . Potts JR, Mokross K, Stouffer PC, Lewis MA (in review) Step selection techniques uncover the environmental predictors of space use patterns in flocks of Amazonian birds. Ecology

23. Hypotheses 1. Birds are more likely to move to higher canopies:

24. Hypotheses 1. Birds are more likely to move to higher canopies: 2. In addition, birds are more likely to move to lower ground: (

25. Maximum likelihood technique 1. Find the that maximises: where and are, respectively, the sequence of positions and trajectories from the data, and

26. Maximum likelihood technique 2. Find the that maximises: where is the value of that maximises the likelihood function on the previous page, and

27. Deriving space use patterns: stochastic simulations Potts JR, Mokross K, Stouffer PC, Lewis MA (in review) Step selection techniques uncover the environmental predictors of space use patterns in flocks of Amazonian birds. Ecology

28. Deriving space use patterns: master equations and PDEs • From the step selection function to a master equation: where is the intersection of with the half-line starting at and continuing on a bearing of . Potts JR, Bastille-Rousseau G, Murray DL, Schaefer JA, Lewis MA (in prep) Predicting local and non-local effects of resources on animal space use using a mechanistic step-selection model

29. Deriving space use patterns: master equations and PDEs • From the step selection function to a master equation: where is the intersection of with the half-line starting at and continuing on a bearing of . • PDE in the simple case where the turning angle distribution is uniform and : Potts JR, Bastille-Rousseau G, Murray DL, Schaefer JA, Lewis MA (in prep) Predicting local and non-local effects of resources on animal space use using a mechanistic step-selection model Moorcroft and Barnett (2008) Mechanistic home range models and resource selection analysis: a reconciliation and unification.Ecology 89(4), 1112–1119

30. Solution of the Moorcroft-Barnett model • Result: where • i.e. the animal’s position distribution at a point is a function of the resource quality at that point

31. Problem: no effect of patch size and isolation Red = better resources Blue = worse resources

32. Simple solution: weight based on start and end of step • H(x), H(y) are the habitats at x and y respectively. • “weight” associated to moving from habitat to .

33. Results: space use in artificial landscape

34. Results: space use in artificial landscape

35. Movement data Statistical tests, e.g. MLE Step selection functions Simulations Space use patterns/data Master equations, PDEs Mathematical analysis

36. Coupled step selection functions One step selection function for each agent and include an interaction term : where represents both the population positions and any traces of their past positions left either in the environment or in the memoryof agent . Potts JR, Mokross K, Stouffer PC, Lewis MA (in prep) A unifying framework for quantifying the nature of animal interactions

37. Amazon birds: testing hypotheses Territorial marking (vocalisations): if any flock is at position at time t otherwise.

38. Amazon birds: testing hypotheses Territorial marking (vocalisations): if any flock is at position at time t otherwise. Hypothesis 1 (tendency not to go into another’s territory):

39. Amazon birds: testing hypotheses Territorial marking (vocalisations): if any flock is at position at time t otherwise. Hypothesis 1 (tendency not to go into another’s territory): Hypothesis 2 (tendency to retreat after visiting another’s territory): where is a von Mises distribution, is the bearing from to and is the bearing from to a central point within the territory and if X is true and 0 otherwise.

40. Amazon birds: space use patterns between competing models

41. Acknowledgements Mark Lewis (UofA) Karl Mokross (Louisiana State) Guillaume Bastille-Rousseau (Trent) Philip Stouffer (Louisiana State) Dennis Murray (Trent) James Schaefer (Trent) Members of the Lewis Lab (UofA)

42. Conclusion Movement and interaction data Statistical tests Coupled step selection functions Simulations “The challenge is to develop a statistical mechanics for ecological systems” Simon Levin The final frontier! Spatial patterns Mathematical analysis

43. Thanks for listening!